United remain in the FA Cup, League Cup and Europa League, and therefore have plenty to play for before the campaign concludes.
However, The Sun report that Solskjaer will not be sacked even if United fail to achieve their objectives this term.
The Norwegian will remain in the Old Trafford hot seat regardless of whether or not they secure a place in Europe next season.
A failure to finish in the top five could see United miss out on the Europa League, although more places will become available if the FA Cup and League Cup are won by clubs that have already qualified for continental competition.
The United board are prepared to give Solskjaer at least two more transfer windows – this month’s and the summer’s – before judging him.
The former Molde manager has now been at the helm for 13 months, although the first three of those were on an interim basis.
Solskjaer signed Aaron Wan-Bissaka, Harry Maguire and Daniel James last summer, and hopes to land Bruno Fernandes before the end of January.
Executive vice-chairman Ed Woodward and other senior figures at Old Trafford acknowledge that it will take time to turn United’s fortunes around.
The club are determined to show patience and have faith that Solskjaer’s long-term strategy will ultimately bear fruit.
After Wednesday’s clash with Burnley, United will face either Tranmere or Watford in the fourth round of the FA Cup this weekend.